Stability threshold of Couette flow for 2D MHD equations in the Sobolev space

Abstract In this paper, we investigate the stability of the 2D Couette flow ( y , 0 ) T under the influence of a uniform magnetic field ( β , 0 ) T . Our focus is on the magnetohydrodynamic (MHD) equations on T × R , characterized by distinct viscosity coefficient ν and magnetic diffusion coefficient µ . We derive space-time estimates for the linearized equations for all β > 0, which capture the enhanced dissipation and inviscid damping effects. As an application, we establish nonlinear stability for perturbations of size c min { ν , μ } 1 2 in a succinct manner.